TOPICS
Search

Schur's Partition Theorem


Schur's partition theorem lets A(n) denote the number of partitions of n into parts congruent to +/-1 (mod 6), B(n) denote the number of partitions of n into distinct parts congruent to +/-1 (mod 3), and C(n) the number of partitions of n into parts that differ by at least 3, with the added constraint that the difference between multiples of three is at least 6. Then A(n)=B(n)=C(n) (Schur 1926; Bressoud 1980; Andrews 1986, p. 53).

The values of A(n)=B(n)=C(n) for n=1, 2, ... are 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, ... (OEIS A003105). For example, for n=15, there are nine partitions satisfying these conditions, as summarized in the following table (Andrews 1986, p. 54).

A(15)=9B(15)=9C(15)=9
13+1+114+115
11+1+1+1+113+214+1
7+7+111+413+2
7+5+1+1+110+512+3
7+1+1+1+...+110+4+111+4
5+5+58+710+5
5+5+1+1+...+18+5+210+4+1
5+1+1+...+18+4+2+19+5+1
1+1+...+17+5+2+18+5+2

The identity A(n)=B(n) can be established using the identity

sum_(n=0)^(infty)B(n)q^n=(-q;q^3)_infty(-q^2;q^3)_infty
(1)
=((q^2;q^6)_infty(q^4;q^6)_infty)/((q;q^3)_infty(q^2;q^3)_infty)
(2)
=((q^2;q^6)_infty(q^4;q^6)_infty)/((q;q^6)_infty(q^4;q^6)_infty(q^2;q^6)_infty(q^5;q^6)_infty)
(3)
=1/((q;q^6)_infty(q^5;q^6)_infty)
(4)
=sum_(n=0)^(infty)A(n)q^n
(5)

(Andrews 1986, p. 54). The identity B(n)=C(n) is significantly trickier.


See also

Andrews-Gordon Identity, Göllnitz's Theorem, Rogers-Ramanujan Identities, Schur's Lemma, Schur Number

Explore with Wolfram|Alpha

References

Andrews, G. E. "q-Series and Schur's Theorem" and "Bressoud's Proof of Schur's Theorem." §6.2-6.3 in q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., pp. 53-58, 1986.Bressoud, D. M. "Combinatorial Proof of Schur's 1926 Partition Theorem." Proc. Amer. Math. Soc. 79, 338-340, 1980.Schur, I. "Über die Kongruenz x^m+y^m=z^m (mod p)." Jahresber. Deutsche Math.-Verein. 25, 114-116, 1916.Schur, I. "Zur additiven Zahlentheorie." Sitzungsber. Preuss. Akad. Wiss. Phys.-Math. Kl., pp. 488-495, 1926. Reprinted in Gesammelte Abhandlungen, Vol. 3. Berlin: Springer-Verlag, pp. 43-50, 1973.Sloane, N. J. A. Sequence A003105/M0254 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Schur's Partition Theorem

Cite this as:

Weisstein, Eric W. "Schur's Partition Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SchursPartitionTheorem.html

Subject classifications